Model Laws (or) Similarity Laws

MODEL LAWS (OR) SIMILARITY LAWS

The results obtained from the model tests can be transferred to the prototype by the use of model laws.

The models are designed on the basis of the force which is dominating in the flow situation.

The laws on which the models are designed for dynamic similarity are called model (or) similarity laws.

In most of the problems only one force is relatively more significant in addition to the inertia force and other forces may not exist or may be insignificant.

Under these circumstances the various model laws have been developed based on the influence of forces different analysis.

The ratio of the corresponding forces acting at the corresponding points in the model and prototype should be equal for the dynamic similarity between the model and the prototype. This means that dimensionless numbers should be same for the model as well as the prototype.

Actually it is very difficult to satisfy the condition that all dimensionless number are same for both model and prototype.

Following are the model laws.

(i) Reynolds model law

(ii) Froude model law

(iii) Euler model ĺaw

(iv) Weber model law

(v) Mach model law

1. Reynolds model law

If the model is designed on the designed on the basis of Reynolds number then the model is said to be Reynolds model law.

Reynolds model law is used when ever inertia and viscous forces are predominant in the fluid as compared to other forces.

This law states that the Reynolds number for the model must be equal to the prototype.

Vm = Velocity of fluid in model.

ρm = Density of fluid in model

Lm = Length (or) linear dimension of the model

μm = Viscosity of fluid in model.

Vp, ρp,  Lp and μp are the corresponding values of shield in prototype.

Applications:

(i) Flow of incompressible fluid in closed pipes, flow meters and fans.

(ii) Flow around the submerged bodies such as aeroplanes and submarines. 

(iii) Flow around structures and other bodies immersed fully in moving fluids 

2. Froude's Model Law:

The law in which models are based on Froude number in called as Froude's model law.

Froude model law is applicable when the gravity force is only predominant force which controls the flow in addition to the force of inertia.

This law states that the Froude number for the model must be equal to prototype. 

(i) Time scale ratio

(ii) Acceleration scale ratio (ar)

(iii) Discharge scale ratio (Qr)

(iv) Force scale ratio (Fr)

(v) Pressure intensity scale ratio (Pr)

(vi) Energy (or) Work scale ratio (Er)

(vii) Momenturm (or) impulse scale ratio (Mr)

Impulse = change in momenturm

M = mv

(viii) Torque scale ratio ((Tr) = Force ratio

Tr = Lr4

(ix) Power scale ratio (Pr)

Applications:

(i) Free surface flows over spillways, sluice etc,

(ii) Flow of jet through orifice or nozzle

(iii) Formation of waves on the surface

(iv) Fluids of different mass densities flowing over one another.

3. Euler's Model Law.

The law in which the models are based on the Euler's number is called as Euler's model law.

Whenever pressure forces are predominant as compare to other forces for dynamic similarity the Euler's numbers in the model must be equal to prototype.

Em = Ep

Euler's law applicable in following cases

(i) Water hammer phenomenon

(ii) Phenomenon of cavitation

(iii) Fully turbulent flow in case closed pipe

(iv) Models of aerofoils, fan blades

(v) Pressure distribution on ship

4. Weber's Model Law.

The law in which the models are based on the Weber's number that is called Weber's model law. When ever surface tension forces are pre-dominant as compare to other forces for dynamic similarity, Weber number of model must be equal to Weber number of prototype.

This law is applicable in following case

(i) Capillary movement of water in soil

(ii) Flow over weights for small heads

(iii) Study of droplet, study of very small jet

(iv) Capillary waves in channels and narrow passages

5. Mach Model Law

In Which law is based on the mach number that is called mach model law, where the elastic Force are pre-dominant as compared to other forces, for dynamic similarity, mach number in the model law must be equal to mach number of the prototype.

This law is applicable for the following cases.

(i) Flow of aeroplane with supersonic speed

(ii) Aero dynamic testing

(iii) Under water testing of torpedo's

(iv) Study of movement of rockets and missiles

(v) Over coming the problem of water hammer analysis